These charts, defined up to an isometry, cover the expanding portion, \(M_+\) , for \(t_c \in (-\infty ,0)\)  and \({\bf x}_c\in {R}^3\)  while the collapsing one, \(M_-\) , is covered by similar charts but with \(t_c >0\)  [1]}. Note that the conformal coordinates for FLRW space-times were introduced by Lemaître [2]}.
